The application of an optimal transport to a preconditioned data matching function for robust waveform inversion

Abstract

Full Waveform Inversion updates the subsurface model iteratively by minimizing a misfit function, which measures the difference between observed and predicted data. The conventional l 2 norm misfit function is widely used as it provides a simple, sample by sample, high resolution misfit function. However it is susceptible to local minima if the low wavenumber components of the initial model are not accurate. A deconvolution of the predicted and observed data offers an extend space comparison, which is more global. The matching filter calculated from the deconvolution has energy focussed at zero lag, like a Dirac Delta function, when the predicted data matches the observed ones. We use the Wasserstein distance to measure the difference between the matching filter and a Dirac Delta function. Unlike data, the matching filter can be easily transformed to a distribution satisfying the requirement of optimal transport theory. Compared with the conventional normalized penalty applied to non-zero lag energy in the matching filter, the new misfit function is a metric and has solid mathematical foundation based on optimal transport theory. Both synthetic and real data examples verified the effectiveness of the proposed misfit function.

Joint Minimization of the Mean and Information Entropy of the Matching Filter Distribution for a Robust Misfit Function in Full-Waveform Inversion

“…If the shift between the predicted and measured signal is larger than a half of a cycle, the l 2 norm would try to match different events, and thus, becomes "cycle-skipped." A remedy to this is to seek global rather than local comparison methods, such as matching filter based approaches [16], [17], [18], [19], [20] or optimal transport misfit functions [21], [22], [23], [19].…”

“…This normalization term has its roots in the instantaneous traveltime objective function as demonstrated by [26]. Based on the optimal transport theory, [18], [20], [27] proposed a more general framework for focusing the resulting matching filter to be a Dirac delta function. In their method, rather than applying a penalty or a normalization term, they formulate the misfit function by measuring the Wasserstein W 2 distance between the resulting matching filter and the Dirac delta function.…”

The effectiveness of a pseudo‐inverse extended Born operator to handle lateral heterogeneity for imaging and velocity analysis applications

“…An accurate macro‐velocity model is essential in pre‐stack depth migration to obtain the correct image and in full waveform inversion (FWI) (Lailly ; Tarantola ) to avoid the cycle skipping for convergence (Alkhalifah ; Sun and Alkhalifah ). Conventional velocity model building tools are based on travel time or ray tracing methods, which utilize the high‐frequency asymptotic approximation and fail to invert for an accurate velocity in complex regions where multi‐arrival and shadow zone occur (Fei et al .…”

Salt Body Inversion Using an Optimal Transport of the Preconditioned Matching Filter

“…The conventional l 2 norm misfit function though simple, and potentially high resolution, is a local comparison of the data and prone to cycle skipping. Thus, many have addressed the cycleskipping problem by introducing more robust misfit functions, which try to compare the data in a more global way such as a matching filter based misfit function [Luo and Sava, 2011,Warner and Guasch, 2016,Sun and Alkhalifah, 2018b or a optimal transport misfit function [Engquist and Froese, 2014,Métivier et al, 2016, Yang et al, 2018, Sun and Alkhalifah, 2018a, Sun and Alkhalifah, 2019.…”